UM HE 73-13 The Scatter Focusing of Multi-GeV Charged Particles and Neutral Hadrons Lawrence W. Jones and Donald G. Koch Randall Laboratory of Physics University of Michigan Ann Arbor, Michigan 48104 May 1973 Abstract The multiple coulomb scattering of charged particles and the nuclear elastic scattering of neutral hadrons may serve as a basis for the design of lenses for beams'of such particles. The optical and spectral properties of such lenses are explored and numerical examples are given.

I. Introduction Charged particles may be focussed by multiple scattering in a suitably designed lens of scattering material. Similarly, neutral hadrons may be focussed by elastic scattering from nuclei using a specially constructed lens. These notions and some experimental observations were originally set forth in papers by one of us in 1964 on "The Random Optics of Particle Beams." In this paper, we present a more complete - and more correct - formulation of the multiple-scattering focusing problem, with numerical examples. We then present a quantitative study of the elastic scattering lens for neutrons (or other neutral hadrons) noting possible applications and numerical examples. The history of this interest began in 1963 when setting up experiments in the extracted proton beam of the Brookhaven Cosmotron. Polaroid film constituted a convenient detector technique for aligning and focussing the beam. In order to locate a reference point or line on the film, a metal object was placed at a known position ahead of the flim and its shadow was recorded on film. The curious properties of these shadows, or (more properly) images, was the starting point for the ideas discussed below. If the film is placed against an object, no image is seen, but if the film is a few cm beyond the object, the image is surprisingly sharp and detailed. That no image is seen in the former case is reasonable, since the attenuation of a 3 GeV proton beam in 1/8 inch of brass is only about 2%/. The image that develops with

-2 -spacing is then due to multiple coulomb scattering in the metal. In order to understand such images, consider a Cartesian coordinate system with a parallel beam in the +z direction and a scattering slab at z=0 extending over all y and all positive x (x>O) with open space at all negative x. Neglecting interactions in the slab, the beam flux at x>>O and at x<<O for positive values of z is unaffected by the slab, since as much flux is scattered into a given area as out of one. On the other hand, for values of x near (xl/zrms (where rms is the multiple coulomb scattering angle), the intensity will be reduced for x positive, since some particles are scattered out of that region, and the intensity will be increased for x negative since no particles are scattered out but some are scattered in. This is illustrated in Figure 1. In Figure 2 we have reproduced the images of two 1-inch diameter, 1-inch long cylinders (axes parallel to the proton beam), one of lead and one of aluminum. The images are positive prints, so that bright areas correspond to higher proton fluxes. That more subtle image detail may be visualized is seen in Figures 3 and 4, where a wrist watch and an electric hand drill are imaged. II. A Multiple Coulomb Scattering Lens The notion that multiple coulomb scattering can be used to enhance the flux of charged particle beam particles at some region downstream of a scatterer suggests that it should be possible to so arrange scatterers as to maximize this flux enhancement, i.e., to design a "lens." We may proceed below with such a lens design, making use of simplified experessions

-3 -2 for multiple coulomb scattering. The probability of scattering through a projected angle 0 upon traversal of a scattering material of thickness t is x given by 1 12 -[ x2/28 2] P(eX)dRx = 1 a e dOx (1) where 15.0/C= (x) (2) go pv R rms' with pv in MeV and R the radiation length in the material. The probability of scattering through a space angle 0 is then -[e2/2e2] P(8)de = 2 e d (3) where Vn 7 ns 21.2 A (4) (8)rms = /7(ex)rms = pv (4) In these expressions and throughout we will use only the small angle approximation where 08 sinG = tane. The probability of scattering into a solid angle dn at 6 is then P(e )Odd = P(e)dO (5) 2w0 2'rre Now consider a uniform parallel flux I of particles (of p,v) incident on a scattering slab of thickness t at z=0 as in Figure 5. At r from the z axis the number of particles incident on a small area dAo will contribute to the flux at z=d, r=O an amount d~ given by

-4 -d o __dAo[ ] dfl (6) where dAd is an element of area normal to the z axis at z=d. Then, in the small angle approximation, dn dAd/d2, r/d 8, and dAo = rdrdco. Therefore, d$ =,o 2 P(e)de. (7) Integrating around an annulus of radius r and width dr over cp from 0 to 2r gives A = ~oP(e)de. (8) If we now use a scatterer in the shape of a disc of inner radius rl and outer radius r2, subtending angles from the "focal" plane (z=d, r'O) e81rl/d, 9r/d, the total flux T in this focal plane is given by 82 T ' l1 + S P(e)d] >o' (9) 1 so that some net gain in flux is achieved. One can do better than use a uniform slab, of course. The scatterer may be contoured in the form of a paraboloid cf revolution such that e - aeo or

-5 -r - 15a A/f (10) d ~pv R' (10) 2 so that t c r. Now 2 2 T =+ J' e-(I /2a a e1, (11) or T o = L1 +.7358 n]. 2/011 This geometry is illustrated in Figure 6. The spectral properties of the lens, or chromatic aberation, can be found by study of Eq.(8), which we may rewrite here as d$(p) = (P)e9de e [ Pi (14) where p is the rms scattering angle for particles of momentum p and h(p) is the flux of those particles. From Eq.(2), P9p = Polo' and from Eq. (12) and (13), T = /9^ 1 so that

-6 -poe 9 = -— * (15) P /7p Consequently 2 2 da(p) = o (p) 2 p — 2 _p2 /~J de d0(p) \= (p) 2(P) eP (16) and $(p) = o,(p) 2 )2 e-[p /p o]n ). (17) The ratio of the flux at p relative to the flux at po, the design momentum, can then be found relative to the incident flux at each momentum. This is: (P^)/ p) E)2 l-(P/Po2' (18) As before, ~T = ~ + $, however it is convenient to leave Eq.(18) in this form as it in independent of 82/81. The response function of Eq.(181 is plotted in Figure 7. Consider a numerical example of such a scatter focusing lens where 82/81 = 20. Here, for the design momentum p, T/o = 3.204. (19) This might in practice correspond to a lens of 20 cm outer diameter with a 1 cm diameter hole in the center. This could be a lead lens of 20 cm diameter and 1/5 of a nuclear interaction mean free path thick at the edge. Since the interaction mean free path in lead is about 14 cm, it would be 2.8 cm thick at r=10 cm. As R =0.51 cm for lead, the focal length would then be found from

-7 -15 35.147 r/f = r/d = 2 = 52.8/.51 = (20) The focal length is then given by pv r2.1 f 15. R/ (21) which, for this numerical example, gives f = 0.2845(pv), where pv is in MeV and f is in cm. Thus for pv = 10GeV, f = 28.45 meters. This lens would enhance the axial flux over the central 1 cm. diameter spot by a factor of 3.2, and with a focal length altogether reasonable. Such a lens is obviously very much less costly than a magnetic lens (quadrupole pair, etc.), it is compact and trouble-free. As such it could be totally buried in shielding, or could be used in an intense radiation environment. On the other hand, the gain through scatter focussing should not be misunderstood. Our factor of 3.2 should properly be compared with a factor of 400 for a properly designed magnetic lens of the same aperture' In fact the gains in scatter-focussing are proportional to tn (aperture) while the gains in proper geometrical optics are proportional to (apertur optics are proportional to (aperture).

-8 -IIIA. Elastic Scatter Focusing of Neutral Hadrons, General Principles The situation is somewhat more interesting in the case of neutral hadrons where no other focusing is possible. Here nuclear diffraction elastic scattering plays a role exactly analogous to multiple coulomb scattering (Section II) except that atomic mass number (nuclear radius) must be used as variable instead of scattering thickness, and scattering must compete with nuclear absorption (inelastic scattering) so that the gains are much more limited. The point is, of course, that no other means exist for performing equivalent operations. Further, the chromatic aberation may be used to bias a continuous energy spectrum to favor particular energies. Our attention is directed primarily toward energies above a GeV. We will use neutrons for our numerical examples, although equivalent expressions almost certainly hold for K beams.3 The expressions which play the role of Eqs.(l)-(5) in the neutron case are given below, where the approximate analytic expressions are taken from the data of Parker et al.4 and Ringia et al. for 4-5 GeV neutrons. The elastic scattering is presumed here to be completely imaginary and due entirely to optical diffraction. In this case, then d T e-Btl T e-Bp22 31 ~~e, (22) dltl 16rh2 16rh2 and since da r d a d= T da Tt '

-9 -2 2 2 da P aT -Bp (23) pd7 a 2= 2 ---5 2 -,(23) dor 2(47T where p is in GeV/c. We will adopt the following approximate but very satisfactory dependences of aT and B on atomic number A: ao =80A2/3mb = 8x10 26A2/3cm2, (24) B = lA2/3(GeV/c)2 If neutrons are incident on a slab of thickness., the maximum number will emerge after experiencing a single scattering if the thickness is equal to the mean free path of neutrons in the material, i.e. 4=X, where X = A/[NpTl1, with A the atomic number, N Avogadro's number, p the density, and a the total cross section. The flux of single scattered nuetrons, Be, is then given in terms of the incident flux, o, by u = $ (Icl (25) s o aT e where a is the elastic scattering cross section. el From Eqs.(22)-(24), Cel CT el =- T = 0.41, (26) aT 167A 2 B so that s/0o = 0.1508. As a result, if we neglect plural scattering, only 15% of the flux incident on a "lens" will be usefpl in scatter focusing. We will explore four possible configurations of neutron lenses.

-11 -2 c 1/B A2/3 82/01 = (A1/A2) 13 (30) A convenient span of materials would range from carbon to lead, where (A1/A2)1/3 =(208/12)1/3 = 2.58. One may do slightly better ranging from beryllium to uranium, with (A1/A2)13 = (238/9)13 = 2.98. The overall enhancement factors are then s_/o = 0.1049, A from 12 to 208 (31) a /o = 0.1208, A from 9 to 238. S 0 In analogy with Eqs. (13) and (19) -T = 1El +.1107Cn(e2/e1), or fT/f o 1.11. (32) This is hardly a spectacular flux improvement. It is however of greater interest here to consider the spectral response of such a lens. Neutron and neutral kaon beams from a multi GeV accelerator are generally produced from a beryllium target in a proton beam. For zero degrees production the neutron spectrum is roughly proportional to p2 up to some momentum close to the proton momentum, beyond which the spectrum rolls off with a tail related to the Fermi motion of nucleons in the target nucleus.6 The momentum spectrum

-12 -enhancement provided by a diffraction-scattering lens would be exactly as represented in Figure 7, and by plugging the axis with several mean free paths of material, a neutron beam would be obtained of about 10% the original intensity and with a spectrum enhanced about a desired momentum. The undesirable aspect of such a lens, in addition to the intensity loss, is the angular divergence of the beam. In many experiments with high energy neutrons the direction of an incident neutron is a necessary constraint in the data. This is normally quite well known from the event vertex and the neutron-producing target. In typical cases pe is the order of 1 to 0.3 MeV-radians. With a carbon scatterer at erms, pe would be 140 MeV-radians for a scatter-focused beam. It should be noted in both the flux and the spectrum calculations here and below that not only plural elastic scattering but all inelastic scattering has been ignored. It has been our experience in studying total cross sections and small angle elastic scattering of neutrons that the neglect of inelastically scattered neutrons over these small angles characteristic of the diffraction region is very appropriate. Charged particles and 7-rays are easily and effectively removed from such beams by sweeping magnets and radiators of high-Z materials. IIIC. The Homogeneous Plate Neutron Lens We will now consider a different "lens" wherein a slab of a single material one interaction mean free path thick containing a small axial hole is used to enhance the neutron flux as in Figure 9. We will refer to this as the "homogeneous plate" case. In this case, Eq. (28) may be integrated with

-13 -9 fixed to give r /2eo2) )(^^Ch = % (0.150)[e 1 e 2. (33) s 0 2 2 2 2 When 02 << 2, >> this becomes simply AS 0.15o' and (34) IT = 1.15. It seems at first surprising that this factor exceeds the factor 1.11 of Eq. (32) for the "optimized" case. The reason, of course, is that the range of 0 here is much greater. On the other hand, no spectral improvement is achieved here. IIID. Combination Lens The third case which may be explored is one wherein a single, high-A medium is used from a minimum aperture to an intermediate radius, then a graded lens of the first type is used from this radius to the radius corresponding to erms of the low-A medium, and the low A medium then extended to a much larger radius. In this case the total gain in flux is indeed the sum of the two, i.e. T= [l + 0.11 + 0.15] = 1.26$. (35) T o o Where maximum flux is the only criterion, this may be an interesting case. The total flux gain, however, seems relatively modest. Such a lens is illustrated diagramatically in Figure 10. IIIE. A Neutron Lens of Three Coaxial Cylinders As a final case we may explore a simple and more realistic configuration in lieu of the lens of IIIB. Nesting cylinders

-14 -of three materials may be used, with one material of high atomic weight A1 extending from ra to rb, a second of intermediate atomic weight A2 from rb to rc, and a third light material of A3 from rc to rd. Each cylinder would be an interaction mean 3 c d free path in length, as illustrated in Figure 11. The ratios of the r's and A's may be chosen to yield the same terms in Eq. (33) for each material. With angles 0 proportional to radii r, we may define intermediate rms scattering angles e1, 82, and 83 as 2 = [lOA12/3p2]-1 2 2 = [0lA22/3p2]-1 22 = [lOA32/3p2-1, so that 1/3 - a1/3 = A1/33 1 1 2 2 3 3* We may set b/ea= e b = ed/ec = e 1/2 = e2/ 3' and eb/ea = (7e1/ea) 2 etc. This determines A2 given A1 and A3 such that A2 =. rl3 3 (36) The scattered flux ~ at the focal plane of the lens is s given in terms of the incident flux o by [r8 -(6 82/2 82) -( eb /281 ) = o (3x0.15)e a e 1, (37) ~s ~o(3x01)

-15 -and the spectral response will be 2r -( a2 -(p2B 8 2 (p) = 0.45 0(p)e e a - (38) Two numerical examples are explored below. First, with A = 207.2 (lead) and A3 = 12.01 (carbon), A2 = 49.88. Titanium, with A = 47.90 is suitably close. Second, with A1 = 238.1 (uranium) and A3 = 9.02 (beryllium), A2 = 46.34, again close to 47.90 (titanium). For the Pb-Ti-C lens, Ia/eb) = (eb/ec) = (8c/8d) = 0.621. (39) From Eq. (37), 0.45[0.621 - e1610 = 0.152t. (40) a 0 0 For the U-Ti-C lens, (6 /eb) = 0.5794, etc. (41) and s = 0.172~. (42) From E. (38), the spectra in the two cases are given by, first, ( ( [e-0621(p/po) -1.61(p/p 2 (P)s^ — ^ Le - e, s (P)/s(po).33752 and, in the second case, (43),p1 [ -0.5794(p/po) 2 -1.726(p/po 2] (p) / {(Po) = i e.38224(44) (44) This spectral response is graphed in Figure 12 together with the spectral response of Eq. (29) of section IIIB (Figure 7). The shape of the response is remarkably similar to that from the structurally impractical continuum of materials.

-16 -It may be interesting to consider a numerical example relevant to the new NAL accelerator. Consider a lens optimized for 300 GeV neutrons, with the relationships from Eqs.(27) and (24), e.g. = f/r = / ~pA1/3/ 7U, etc. 1/0a f/ra 5 1 b a The parameters for a lens of focal length f = 100 m are noted in Table I. Table I Lenses of 100 m focal length optimized for po = 300 GeV/c neutrons. Material Length (cm) Radius (cm) s/o -—............ ii _ i tli 1.404 ] Pb 9.5 2.260 Ti 16.8 0.152.639 C 23.8 5.858 1.294 U 6.9 2.234 Ti 16.8 0.172 3.856 Be 24.0 6.655 At NAL there is a neutron beam brought to detectors about 400 m from the neutron production target. By locating such a lens at 200 m from the target and plugging the central hole with an iron rod absorber of at least one meter, and placing a collimator of one or two cm aperture at 400 m, an enriched flux of about 15% of the original beam (through that aperture) would be achieved.

-17 -IIIF. Two-Stage Neutron Focusing It was suggested by H.R. Gustafson that in some cases the neutron flux is much greater than needed and that the shape of the spectrum is all important, so that it might be desirable to use a second lens in tandem with the first. The flux at the focus of the second lens would now be about 15% of the flux incident on the second lens, if it were incident as a parallel beam. In fact, the flux is diverging from the first focus, so that if the second lens is half way between the first and the second foci, the final flux is reduced an additional factor of four. The situation is sketched in Figure 13. The number of neutrons at the second focus N2 is given by 2 = $ A = 0.152 $ 'A 2 = ~2A2 4 o 2 where $ ' is the flux incident on the second lens. If N1 is the number of neutrons transmitted through the first collimator, A1 is the area of the first collimator, and A0 the area of the lens, N, = A, = $'At, N1 = sA1 0 o, so that 0N.152 A1A2 (0.152) 2 1 2 4 s A 4 o A or 3A. ~2 [5.8x10-3 A] lo For a 2 cm diameter aperture A1 and the Pb-Ti-C lens of Table I, =2 s 1.79xl10 4.

-18 -This assumes both lenses contain plugs to block the axial beam, and applies only to the "design" momentum p. The resulting spectral response, which is achieved at the cost of this loss, is the square of the curve of Figure 12 and is plotted in Figure 14. It appears to us that only rarely will such a loss in flux justify the spectral response gain. IV. Conclusions We have shown how very simple multiple scattering lenses may provide flux enhancements of charge particles by modest factors (x2 - x4). The practical utilization of such lenses may be very limited except for such special applications as neutrino beams. Multiple coulomb scattering also provides the basis for a curious form of radioqraphy using multi-GeV charged particles. Here discontinuities in scattering properties are visualized yielding results somewhat analogous to Xerographic X-rays. Utilizing similar mathematics, the nuclear elastic scattering of multi-GeV neutrons and other neutral hadrons serves as a mechanism for focusing such particles. The absolute flux gain thus achieved in neutral hadron beams is less than 50%, however the chromatic aberation of such lenses may serve to provide a differential flux enhancement such that a spectrum may be peaked about a desired momentum. A two-stage lens system is possible, although the greater spectral improvement is only achieved with severe loss of flux.

-19 -References 1. "The Random Optics of Particle Beams," L.W. Jones, Univ. of Michigan Technical Report 03106-15-T (1964) (unpublished). L.W. Jones, p. 569, XII International Conference on High Energy Physics, Vol. 2. Atomizdat (Moscow) (1966). 2. B. Rossi, "High Energy Particles," p. 66, (Prentice Hall) (1952). W.H. Barkas and A.H. Rosenfeld, UCRL-8030 (1963) TID-4500 (unpublished). 3. W.L. Lakin, E.B. Hughes, L.H. O'Neill, J.N. Otis, and L. Madansky, Phys. Lett. 31B, 677 (1970). 4. E.F. Parker, T. Dobrowolski, H.R. Gustafson, L.W. Jones, M.J. Longo, F.E. Ringia, and B. Cork, Phys. Lett. 31B, 246 (1970). 5. F.E. Ringia, T. Dobrowolski, H.R. Gustafson, L.W. Jones, M.J. Longo, E.F. Parker, and B. Cork, Phys. Rev. Lett. 28, 185 (1972). 6. B.G. Gibbard, thesis, Report UM-HE-70-11 Randall Laboratory of Physics, University of Michigan (1970) (unpublished). T.P. McCorriston, Jr., thesis, Report UM-HE-72-11 Randall Laboratory of Physics, University of Michigan (1972) (unpublished). D.D. O'Brien,, thesis, Report UM —72-32 Randall Laboratory of Physics, University of Michigan (1972) (unpublished).

(8) // ~SCATTERING OBJECT (a) __________ AT = 0 EXTENDING — x FROM X=O TO POSITIVE X (b) BEAM INTENSITY m AT = +i % 0 (BEAM OF PARTICLES MOVING IN THE + i DIRECTION) Fig. 1. The nature of an image resulting from a semiinfinite slab of scattering material. (a) The scatterer, at z=0; (b) the particle flux { vs. x at z >0. Fig. 2. Scatter images of cylinders of lead (left) and aluminum (right) in a proton beam of 3 GeV. Each cylinder was 1 inch in diameter and 1 inch in length. The film was 15 inches beyond the cylinders. The images are positive prints, i.e. lighter shades are higher intensity.

Fig. 3. A wrist watch imaged by scattering of 3 GeV protons. i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~: ~~~~~-~~~~~~~~~~i~~~~~~~i~~~~~~~a-:i::::::ji::::...~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...... Fig. 4. A portion of an electric hand drill imaged with 3 GeVsU p-rotons.n -

dAo bo -;S 4 Sb_ 0 i XQI --- —-^-, ^ ^dAd =I I', 3=D Fig. 5. Scattering of a parallel beam by a plane slab of thickness A. PARALLEL BEAM OF INCIDENT =0 PARTICLES i ' t r ~~ > T | rX 0bT d " \___ _____~ ~(FOCUS) MULTIPLE-SCATTERING LENS Fig. 6. The geometry of a multiple scattering lens, with a thickness 2<r2 from r1 to r2.

u) C):j rci cJ =.//~~~L~ H U) 44 r -q - /0,./0~~~~~~~~r 4*d.""~~~~~~~ —,- r] 4)) p //CVI ~ r 44 o a) ( ci C Om.^-0 rd 0 -\ u W \O 44 00 U o o \ 0 U)* ' O O /o t! 0 0 CO c0 A ~0 CN4 O ^rCt 4.. T —~ ~ ~~~~~~~~~~~~~~~~ N ci)o g

dr Fig. 8. Neutron scattering annulus of length I=X made of materiil with atomic number A. ^ -{J Fig. 9. Homogeneous plate neutron scatterer extending from r1 to r2 (subtending angles e1 and 92 from the focal point).

?, RING OF LOW A U~S \ RINGS OF DIFFERENT A, r o.r A-1/3 _ } RING OF HIGH A =~.*, '...... Fig. 10. Schematic section of combination lens with high atomic number medium inside and with low atomic number medium outside a region in which the atomic number is graded according to rA 1/3. " \ \ \ A3 TC ---. b ra _.-A X' Fig. ll. Schematic section of three-element neutron lens of materials of atomic weights A1, A2, and A3.

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